$$\phi :\mathbb R^3 \rightarrow \mathbb R^6$$ $$(u,v,w)\rightarrow \phi(u,v,w)=(x_1,x_2,x_3,x_4,x_5,x_6)$$
where $ \quad x_1=u^2 \quad x_2=v^2 \quad x_3=w^2 \quad x_4=vw \quad x_5=uw \quad x_2=uv$
For the Jacobian Matrix I get the following:
\begin{pmatrix}
2u & 0 & 0 \\
0 & 2v & 0 \\
0 & 0 & 2w \\
0 & w & v \\
w & 0 & u \\
v & u & 0 \\
\end{pmatrix}
I am looking for subsets $\Omega \subset R^3 \; $ so that $\; \phi$ is an Immersion. Additionally, what are the largest convex subset for which $\phi$ is an Immersion. For which convex subsets is $\phi$ injective and $\phi^{-1}$ continous ?
$\phi \; $ is clearly $C^{\infty}\;$ and the Rank of the Jacobian Matrix is 3 for any open subset $\Omega \subset \mathbb R^3-(0,0,0)$. An open convex subset would be $\mathbb R_+^3:=\{(x,y,z): x,y,z>0 \}$
Am I right ? If so can someone give me a hint on how to find a convex subset for which $\phi \;$ is injective ? Is $\phi^{-1}\;$continous for any of those subsets ?